Algorithm for Testing the Leibniz Algebra Structure

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1 Algorithm for Testing the Leibniz Algebra Structure José Manuel Casas 1, Manuel A. Insua 2, Manuel Ladra 2, and Susana Ladra 3 1 Universidad de Vigo, Pontevedra, Spain jmcasas@uvigo.es 2 Universidad de Santiago, E Santiago, Spain avelino.insua@gmail.com, manuel.ladra@usc.es 3 Universidad de A Coruña, A Coruña, Spain sladra@udc.es Abstract. Given a basis of a vector space V over a field K and a multiplication table which defines a bilinear map on V, we develop a computer program on Mathematica which checks if the bilinear map satisfies the Leibniz identity, that is, if the multiplication table endows V with a Leibniz algebra structure. In case of a positive answer, the program informs whether the structure corresponds to a Lie algebra or not, that is, if the bilinear map is skew-symmetric or not. The algorithm is based on the computation of a Gröbner basis of an ideal, which is employed in the construction of the universal enveloping algebra of a Leibniz algebra. Finally, we describe a program in the NCAlgebra package which permits the construction of Gröbner bases in non commutative algebras. 1 Introduction A classical problem in Lie algebras theory is to know how many different (up to isomorphisms) finite-dimensional Lie algebras are for each dimension [11,15]. The classical methods to obtain the classifications essentially solve the system of equations given by the bracket laws, that is, for a Lie algebra g over a field K with basis {a 1,...,a n }, the bracket is completely determined by the scalars c k ij K such that n [a i,a j ]= c k ij a k (1) so that the Lie algebra structure is determined by means of the computation of the structure constants c k ij. In order to reduce the system given by (1) for i, j {1, 2,..., n}, different invariants as center, derived algebra, nilindex, nilradical, Levi subalgebra, Cartan subalgebra, etc., are used. Nevertheless, a new approach by using Gröbner bases techniques is available [9,10]. On the other hand, in 1993, J.-L. Loday [17] introduced a non-skew symmetric generalization of Lie algebras, the so called Leibniz algebras. They are K-vector k=1 R. Moreno-Díaz et al. (Eds.): EUROCAST 2009, LNCS 5717, pp , c Springer-Verlag Berlin Heidelberg 2009

2 178 J.M. Casas et al. spaces g endowed with a bilinear map [, ] :g g g such that the Leibniz relation holds [x, [y, z]] = [[x, y],z] [[x, z],y], for all x, y, z g. (2) When the bracket satisfies [x, x] = 0 for all x g, then the Leibniz identity (2) becomes the Jacobi identity, and so a Leibniz algebra is a Lie algebra. From the beginning, the classification problem of finite-dimensional Leibniz algebras is present in a lot of papers [1,2,3,4,5,6,8]. Nevertheless, the space of solutions of the system given by the structure constants becomes very hard to compute, especially for dimensions greater than 3 because the system has new equations coming from the non-skew symmetry of the bracket. In these situations, the literature only collects the classification of specific classes of algebras (solvable, nilpotent, filiform, etc.). In order to simplify the problem, new techniques applying Gröbner bases methods are developed [14]. However, the space of solutions is usually huge and two kind of problems can occur in applications: 1. Given a multiplication table of an algebra for a fixed dimension n, how can we know if it corresponds to a Leibniz algebra structure among the ones obtained by the classification? 2. Since the classification provides a lot of isomorphism classes, how can we be sure that the classification is well done? The aim of the present paper is to give an answer to the following question: Given a multiplication table of an algebra for a fixed dimension n, howcan we know if it corresponds to a Leibniz algebra structure? We present a computer program in NCAlgebra [12] (a package running under Mathematica which permits the construction of Gröbner bases in non commutative algebras) that implements the algorithm to test if a multiplication table for a fixed dimension n corresponds to a Leibniz algebra structure. In case of a positive answer, the program distinguishes between Lie and non-lie algebras. The algorithm is based on the computation of a Gröbner basis of an ideal, which is employed in the construction of the universal enveloping algebra UL(g) [19] of a Leibniz algebra g over a field K with basis {a 1,a 2,...,a n }. In order to do this, we consider the ideal J =< { Φ(r [ai,a j])+x i x j x j x i, Φ(l [ai,a j])+y i x j x j y i,i,j =1,...,n} > of the free associative non-commutative unitary algebra K < y 1,...,y n,x 1,...,x n >,whereφ : T (g l g r ) K < y 1,...,y n,x 1,...,x n > is the isomorphism given by Φ(l ai )=y i,φ(r ai )=x i, being g l and g r two copies of g, and an element x g corresponds to the elements l x and r x in the left and right copies, respectively. Then (g, [, ]) is a Leibniz algebra if and only if the Gröbner basis corresponding to the ideal J with respect to any monomial order does not contain linear polynomials in the variables y 1,...,y n. We also obtain that (g, [, ]) is a Lie algebra if and only if the Gröbner basis with respect to any monomial order of the ideal J =< { Φ(r [ai,a j]) +x i x j x j x i,i,j = 1,...,n} > does not contain linear polynomials in the variables x 1,...,x n.

3 2 On Leibniz Algebras Algorithm for Testing the Leibniz Algebra Structure 179 Definition 1. A Leibniz algebra g is a K-vector space equipped with a bilinear map [, ] :g g g satisfying the Leibniz identity [x, [y, z]] = [[x, y],z] [[x, z],y], for all x, y, z g. (2) When the bracket satisfies [x, x] = 0 for all x g, then the Leibniz identity (2) becomes the Jacobi identity; so a Leibniz algebra is a Lie algebra. Hence, there is a canonical inclusion functor from the category Lie of Lie algebras to the category Leib of Leibniz algebras. This functor has as left adjoint the Liezation functor which assigns to a Leibniz algebra g the Lie algebra g Lie = g/g ann,where g ann = {[x, x],x g}. Example Lie algebras. 2. Let A be a K-associative algebra equipped with a K-linear map D : A A satisfying D(a(Db)) = DaDb = D((Da)b), for all a, b A. (3) Then A with the bracket [a, b] =adb Dba is a Leibniz algebra. If D = Id, we obtain the Lie algebra structure associated to an associative algebra. If D is an idempotent algebra endomorphism (D 2 = D) ord is a derivation of square zero (D 2 = 0), then D satisfies equation (3) and the bracket gives rise to a structure of non-lie Leibniz algebra. 3. Let D be a dialgebra [18]. Then (D, [, ]) is a Leibniz algebra with respect to the bracket defined by [x, y] =x y y x, x, y D. 4. Let g be a differential Lie algebra, then (g, [, ] d )with[x, y] d := [x, dy] is a non-lie Leibniz algebra. For a Leibniz algebra g, we consider two copies of g, left and right, denoted by g l and g r, respectively. For an element x g, wedenotebyl x and r x the corresponding elements in the left and right copies, respectively. The universal enveloping algebra of g wasdefinedin[19]as UL(g) :=T (g l g r )/I where T (V ) is the tensor algebra on V and I is the two-sided ideal spanned by the following relations: i) r [x,y] (r x r y r y r x ) ii) l [x,y] (l x r y r y l x ) iii) (r y + l y )l x. Moreover, [19, Theorem (2.3)] establishes that the category of representations (resp. co-representations) of the Leibniz algebra g is equivalent to the category of right (resp. left) modules over UL(g). After this, [16, Corollary 1.4] establishes

4 180 J.M. Casas et al. the equivalence between the categories of representations and co-representations of a Leibniz algebra. Let g be a finite-dimensional Leibniz algebra with basis {a 1,...,a n }.There is an isomorphism of algebras Φ : T (g l g r ) K <y 1,...,y n,x 1,...,x n > given by Φ(l ai )=y i,φ(r ai )=x i,wherek <y 1,...,y n,x 1,...,x n > denotes the free associative non-commutative unitary K-algebra of polynomials. Having in mind the inclusion g T (g l g r ) and the isomorphism Φ, we obtain that UL(g) K <y 1,...,y n,x 1,...,x n > = < Φ(r [ai,a j])+x i x j x j x i, Φ(l [ai,a j])+y i x j x j y i, (x i + y i )y j >, and hence we can use the theory of Gröbner bases on K <y 1,...,y n,x 1,...,x n > [20] to obtain results about the structure of g. Let be a given monomial order on the noncommutative polynomial ring K<X>. For an arbitrary polynomial p K<X>, we will use lm(p) todenote the leading monomial of p. Definition 2. Let I be a two-sided ideal of K<X>. A subset {0} G I is called a Gröbner basis for I if for every 0 f I, thereexistsg G, such that lm(g) is a factor of lm(f). Lemma 1 (Diamond Lemma [7]). Let be a monomial order on K<X> and let G = {g 1,...,g m } be a set of generators of an ideal I in K<X>. Ifall the overlap relations involving members of G reduce to zero modulo G, then G is a Gröbner basis for I. We note that the overlap relations are the noncommutative version of the S- polynomials. Now, we consider the ideal J =< {g ij = Φ(r [ai,a j]) +x i x j x j x i,h ij = Φ(l [ai,a j])+y i x j x j y i,i,j =1,...,n} >. Fori, j, k {1,...,n}, letbep ijk = h ij x k + y i g kj. Lemma 2. P ijk J Φ(l [[ai,a j],a k ]) Φ(l [ai,[a k,a j]])+φ(l [[ai,a k ],a j]) for all i, j, k {1,...,n}. Proof. P ijk = h ij x k +y i g kj = Φ(l [ai,a j])x k x j y i x k y i Φ(r [ak,a j])+y i x k x j J Φ(l [ai,a j])x k x j y i x k y i Φ(r [ak,a j])+φ(l [ai,a k ])x j +x k y i x j J Φ(l [ai,a j])x k y i Φ(r [ak,a j])+φ(l [ai,a k ])x j + x k y i x j x j Φ(l [ai,a k ]) x j x k y i J Φ(l [ai,a j])x k y i Φ(r [ak,a j]) +Φ(l [ai,a k ])x j + x k y i x j x j Φ(l [ai,a k ])+Φ(r [ak,a j])y i x k x j y i J Φ(l [ai,a j])x k y i Φ(r [ak,a j]) + Φ(l [ai,a k ])x j + x k Φ(l [ai,a j]) x j Φ(l [ai,a k ]) + Φ(r [ak,a j])y i. Let [a k,a j ]=α kj 1 a α kj n a n. Φ(r [ak,a j])y i y i Φ(r [ak,a j]) =(α kj 1 x 1+ +α kj n x n )y i y i (α kj 1 x 1+ +α kj n x n )= α kj 1 (x 1y i y i x 1 )+ +α kj n (x n y i y i x n ) J α kj 1 Φ(l [a i,a 1]) α kj n Φ(l [ai,a n]) = Φ(l [ai,[a k,a j]]).

5 Algorithm for Testing the Leibniz Algebra Structure 181 Φ(l [ai,a j])x k x k Φ(l [ai,a j]) =(α ij 1 y 1+ +α ij n y n)x k x k (α ij 1 y 1+ +α ij n y n)= α ij 1 (y 1x k x k y 1 )+ +α ij n (y nx k x k y n ) J α ij 1 Φ(l [a 1,a k ])+ +α ij n Φ(l [a n,a k ])= Φ(l [[ai,a j],a k ]). Theorem 1. Let g be a finite-dimensional K-vector space with basis {a 1,...,a n } together with a bilinear map [, ] :g g g. (g, [, ]) is a Leibniz algebra if and only if the Gröbner basis G = {g 1,...,g t } corresponding to the ideal J with respect to any monomial order does not contain linear polynomials g j (y 1,...,y n ). Proof. If (g, [, ]) is a Leibniz algebra, then P ijk =0foralli, j, k {1,...,n} because of the Leibniz identity (2). On the other hand, if (g, [, ]) is not a Leibniz algebra, then there exist i, j, k {1,...,n} such that [[a i,a j ],a k ]+[a i, [a k,a j ]] [[a i,a k ],a j ] 0,soJ contains a degree 1 polynomial on the variables y 1,...,y n. Example 2. Let (g =< a 1,a 2,a 3 > K, [, ]) be the vector space such that [a 1,a 3 ]=a 2,[a 2,a 3 ]=a 1 + a 2,[a 3,a 3 ]=a 3, and 0 in other case. In this case, J =< x 1 x 2 + x 2 x 1, x 1 x 3 + x 3 x 1 + x 2,x 1 x 2 x 2 x 1, x 2 x 3 + x 3 x 2 + x 1 + x 2,x 1 x 3 x 3 x 1,x 2 x 3 x 3 x 2,x 3,x 1 y 1 y 1 x 1,x 2 y 1 y 1 x 2,x 3 y 1 y 1 x 3 + y 2,x 1 y 2 y 2 x 1,x 2 y 2 y 2 x 2,x 3 y 2 y 2 x 3 + y 1 + y 2,x 1 y 3 y 3 x 1,x 2 y 3 y 3 x 2,x 3 y 3 y 3 x 3 + y 3 > K<y 1,y 2,y 3,x 1,x 2,x 3 >. The Gröbner basis of J with respect to degree lexicographical ordering with y 3 >y 2 >y 1 >x 3 >x 2 >x 1 is {y 3,y 2,y 1,x 3,x 2,x 1 }. Therefore, g is not a Leibniz algebra. According to Theorem 1, we can know if a finite-dimensional algebra is Leibniz or not. In case of positive answer, it is natural to ask if the Leibniz algebra is a Lie algebra or a non-lie Leibniz algebra. As it is well-known [13] g ann =< {[a i,a i ],i=1,...,n} {[a i,a j ]+[a j,a i ],i,j =1,...,n} >, and having in mind the identification g T (g l g r ) a i r ai x i then the generators of g ann can be written as Φ K <x 1,...,x n,y 1,...,y n > {Φ(r [ai,a i]),i=1,...,n} {Φ(r [ai,a j])+φ(r [aj,a i]), i,j =1,...,n} and these generators belongs to J, since g ii = Φ(r [ai,a i]), i =1,...,n, g ij g ji = Φ(r [ai,a j])+φ(r [aj,a i]), i,j =1,...,n. Consequently, if we compute the Gröbner basis G with respect to any monomial order for the ideal J, we can obtain two possible results:

6 182 J.M. Casas et al. 1. There are some linear polynomials g j (y 1,...,y n )ing, sog is not a Leibniz algebra. 2. There are not any linear polynomials g j (y 1,...,y n )ing, sog is a Leibniz algebra, and: (a) If there are some linear polynomials g i (x 1,...,x n )ing, theng ann 0 and so g is a non Lie-Leibniz algebra. (b) In other case g is a Lie algebra. Example 3. Let (g =< a 1,a 2,a 3 > K, [, ]) be the vector space such that [a 1,a 3 ]=a 2,[a 2,a 3 ]=a 1 + a 2,[a 3,a 3 ]=a 1, and 0 in other case. In this case, J =< x 1 x 2 + x 2 x 1, x 1 x 3 + x 3 x 1 + x 2,x 1 x 2 x 2 x 1, x 2 x 3 + x 3 x 2 + x 1 + x 2,x 1 x 3 x 3 x 1,x 2 x 3 x 3 x 2,x 1,x 1 y 1 y 1 x 1,x 2 y 1 y 1 x 2,x 3 y 1 y 1 x 3 + y 2,x 1 y 2 y 2 x 1,x 2 y 2 y 2 x 2,x 3 y 2 y 2 x 3 + y 1 + y 2,x 1 y 3 y 3 x 1,x 2 y 3 y 3 x 2,x 3 y 3 y 3 x 3 + y 1 > K<y 1,y 2,y 3,x 1,x 2,x 3 >. The Gröbner basis of J with respect to degree lexicographical ordering with y 3 >y 2 >y 1 >x 3 >x 2 >x 1 is {x 2,x 1, x 3 y 1 + y 1 x 3 y 2, x 3 y 2 + y 2 x 3 y 1 y 2, x 3 y 3 + y 3 x 3 y 1 }. Therefore, g is a Leibniz algebra and not a Lie algebra. Example 4. Let (g =< a 1,a 2,a 3,a 4 > K, [, ]) be the vector space such that [a 4,a 1 ]=a 1, [a 1,a 4 ]= a 1, [a 4,a 2 ]=a 2, [a 2,a 4 ]= a 2, [a 4,a 3 ]=a 3, [a 3,a 4 ]= a 3, and 0 in other case. In this case, J =< x 1 x 2 + x 2 x 1, x 1 x 3 + x 3 x 1, x 1 x 4 + x 4 x 1 x 1,x 1 x 2 x 2 x 1, x 2 x 3 +x 3 x 2, x 2 x 4 +x 4 x 2 x 2,x 1 x 3 x 3 x 1,x 2 x 3 x 3 x 2, x 3 x 4 +x 4 x 3 x 3,x 1 x 4 x 4 x 1 + x 1,x 2 x 4 x 4 x 2 + x 2,x 3 x 4 x 4 x 3 + x 3,x 1 y 1 y 1 x 1,x 2 y 1 y 1 x 2,x 3 y 1 y 1 x 3,x 4 y 1 y 1 x 4 y 1,x 1 y 2 y 2 x 1,x 2 y 2 y 2 x 2,x 3 y 2 y 2 x 3,x 4 y 2 y 2 x 4 y 2,x 1 y 3 y 3 x 1,x 2 y 3 y 3 x 2,x 3 y 3 y 3 x 3,x 4 y 3 y 3 x 4 y 3,x 1 y 4 y 4 x 1 +y 1,x 2 y 4 y 4 x 2 + y 2,x 3 y 4 y 4 x 3 + y 3,x 4 y 4 y 4 x 4 > K<y 1,y 2,y 3,y 4,x 1,x 2,x 3,x 4 >. The Gröbner basis of J with respect to degree lexicographical ordering with y 4 > y 3 > y 2 > y 1 > x 4 > x 3 > x 2 > x 1 is { x 1 x 2 + x 2 x 1, x 1 x 3 + x 3 x 1, x 1 x 4 + x 4 x 1 x 1, x 2 x 3 + x 3 x 2, x 2 x 4 + x 4 x 2 x 2, x 3 x 4 + x 4 x 3 x 3, x 1 y 1 +y 1 x 1, x 2 y 1 +y 1 x 2, x 3 y 1 +y 1 x 3, x 4 y 1 +y 1 x 4 +y 1, x 1 y 2 +y 2 x 1, x 2 y 2 +y 2 x 2, x 3 y 2 +y 2 x 3, x 4 y 2 +y 2 x 4 +y 2, x 1 y 3 +y 3 x 1, x 2 y 3 +y 3 x 2, x 3 y 3 + y 3 x 3, x 4 y 3 + y 3 x 4 + y 3, x 1 y 4 + y 4 x 1 y 1, x 2 y 4 + y 4 x 2 y 2, x 3 y 4 + y 4 x 3 y 3, x 4 y 4 + y 4 x 4 }. Therefore, g is a Lie algebra. We can also solve the dichotomy of knowing if a K-vector space with a given multiplication table is a Lie algebra by means of the following ideal J =< {g ij = Φ(r [ai,a j])+x i x j x j x i,i,j =1,...,n} >. For i, j, k {1,...,n}, letbeq ijk = x j g ki g ij x k. Lemma 3. Q ijk J Φ(r [ak,[a i,a j]]) +Φ(r [[ak,a i],a j]) Φ(r [[ak,a j],a i]) for all i, j, k {1,...,n}.

7 Algorithm for Testing the Leibniz Algebra Structure 183 Proof. x j g ki g ij x k = Φ(r [ai,a j])x k x i x j x k x j Φ(r [ak,a i]) +x j x k x i gkj x i Φ(r [ai,a j])x k x i x j x k x j Φ(r [ak,a i]) Φ(r [ak,a j])x i + x k x j x i xig kj Φ(r [ai,a j])x k x j Φ(r [ak,a i]) Φ(r [ak,a j])x i +x k x j x i +x i Φ(r [ak,a j]) x i x k x j gki x j Φ(r [ai,a j])x k x j Φ(r [ak,a i]) Φ(r [ak,a j])x i + x k x j x i + x i Φ(r [ak,a j]) +Φ(r [ak,a i])x j x k x i x j = Φ(r [ai,a j])x k + x k (x j x i x i x j )+Φ(r [ak,a i])x j x j Φ(r [ak,a i]) Φ(r [ak,a j])x i + x i Φ(r [ak,a j]) J Φ(r [ai,a j])x k x k Φ(r [ai,a j]) +Φ(r [ak,a i])x j x j Φ(r [ak,a i]) Φ(r [ak,a j])x i + x i Φ(r [ak,a j]). Let [a i,a j ]=α ij 1 a α ij n a n. Φ(r [ai,a j])x k x k Φ(r [ai,a j]) =(α ij 1 x 1+ +α ij n x n )x k x k (α ij 1 x 1+ +α ij n x n )= α ij 1 (x 1x k x k x 1 )+ + α ij k 1 (x k 1x k x k x k 1 )+α ij k 0+αij k+1 (x k+1x k x k x k+1 )+ + α ij n (x nx k x k x n ) J α ij 1 Φ(r [a 1,a k ])+ + α ij k 1 Φ(r [a k 1,a k ]) α ij k+1 Φ(r [a k,a k+1 ]) α ij n Φ(r [ak,a n]) J α ij 1 Φ(r [a 1,a k ])+...+α ij k 1 Φ(r [a k 1,a k ])+ α ij k+1 Φ(r [a k+1,a k ]) α ij n Φ(r [an,a k ]) = Φ(r [[ai,a j],a k ]) α ij k Φ(r [a k,a k ]) J Φ(r [[ai,a j],a k ]), or Φ(r [ai,a j])x k x k Φ(r [ai,a j]) J Φ(r [ak,[a i,a j]]). Theorem 2. Let g be a finite-dimensional K-vector space with basis {a 1,..., a n } together with a bilinear map [, ] :g g g. (g, [, ]) is a Lie algebra if and only if the Gröbner basis G = {g 1,...,g t } corresponding to the ideal J with respect to any monomial order does not contain linear polynomials g j (x 1,...,x n ). Example 5. Let (g =< a 1,a 2,a 3,a 4 > K, [, ]) be the vector space such that [a 4,a 1 ]=a 2, [a 1,a 4 ]= a 2, [a 4,a 2 ]=a 3, [a 2,a 4 ]= a 3, [a 4,a 3 ]=a 1 + a 2, [a 3,a 4 ]= a 1 a 2, and 0 in other case. In this case, J =< x 1 x 2 + x 2 x 1, x 1 x 3 + x 3 x 1, x 1 x 4 + x 4 x 1 x 2,x 1 x 2 x 2 x 1, x 2 x 3 +x 3 x 2, x 2 x 4 +x 4 x 2 x 3,x 1 x 3 x 3 x 1,x 2 x 3 x 3 x 2, x 3 x 4 +x 4 x 3 x 1 x 2,x 1 x 4 x 4 x 1 +x 2,x 2 x 4 x 4 x 2 +x 3,x 3 x 4 x 4 x 3 +x 1 +x 2 > K<x 1,x 2,x 3, x 4 >. The Gröbner basis of J with respect to degree lexicographical ordering with x 4 >x 3 >x 2 >x 1 is { x 1 x 2 + x 2 x 1, x 1 x 3 + x 3 x 1, x 1 x 4 + x 4 x 1 x 2, x 2 x 3 + x 3 x 2, x 2 x 4 + x 4 x 2 x 3, x 3 x 4 + x 4 x 3 x 1 x 2 }. Therefore, g is a Lie algebra. 3 Computer Program In this section we describe a program in NCAlgebra [12] (a package running under Mathematica) that implements the algorithms discussed in the previous section. The program computes the reduced Gröbner basis of the ideal J which determines if the introduced multiplication table of g corresponds to a Leibniz algebra or not. In case of positive answer, the program decides whether the algebra is a Lie or non-lie Leibniz algebra. The Mathematica code together with some examples are available in

8 184 J.M. Casas et al. ##################################################################### (* This program tests if an introduced multiplication table corresponds to a Lie algebra, a non-lie Leibniz algebra or an algebra which has not Leibniz algebra structure. To run this code properly it is necessary to load the NCGB package*) ##################################################################### (* Let g =< a 1,...,a n > be an algebra of dimension n *) (* Insert the Bracket represented by Bracket[{i1,i2}]:={λ 1,...,λ n } where [a i1,a i2 ]=λ 1 a 1 + +λ n a n. In Example 2, e.g., Bracket[{2, 3}] := {1,1,0} *) LeibnizQ[n_]:=Module[{G,A,lengA,Variabs,BaseG,varaux,rvarsx,lvarsy}, (* First of all we construct the generators of the ideal *) G={}; A=Tuples[Table[i,{i,1,n}],2]; lenga=length[a]; Do[ G=Join[ G,{Bracket[A[[i]]].Table[x[i],{i,1,n}]-(x[A[[i,1]]]** x[a[[i,2]]]- x[a[[i,2]]]**x[a[[i,1]]])}],{i,1,lenga}]; Do[ G=Join[ G,{Bracket[A[[i]]].Table[y[i],{i,1,n}]-(y[A[[i,1]]]** x[a[[i,2]]]- x[a[[i,2]]]**y[a[[i,1]]])}],{i,1,lenga}]; rvarsx=table[x[i],{i,1,n}]; lvarsy=table[y[i],{i,1,n}]; Variabs=Join[rvarsx,lvarsy]; (* Now we compute a Gröbner Basis of G *) SetNonCommutative@@Variabs; SetMonomialOrder[Variabs] ; BaseG=NCMakeGB[G,15]; (* Finally, we check the Gröbner Basis *) varaux=select[baseg,union[variables[#],lvarsy]===lvarsy&];

9 Algorithm for Testing the Leibniz Algebra Structure 185 If[varaux==={}, (* g is a Leibniz algebra but, is g a Lie algebra? *) varaux=select[baseg,union[variables[#],rvarsx]===rvarsx&]; If[varaux==={}, Print["g is a Lie algebra"];, Print["g is not a Lie algebra but g is a Leibniz algebra"]; ];, Print["g is not a Leibniz algebra"]; ]; ] Acknowledgements First and third authors were supported by Ministerio de Educación y Ciencia, Grant MTM C02-02 (European FEDER support included) and by Xunta de Galicia, Grant PGIDIT06PXIB371128PR. References 1. Albeverio, S., Ayupov, S.A., Omirov, B.A.: On nilpotent and simple Leibniz algebras. Comm. Algebra 33(1), (2005) 2. Albeverio, S., Omirov, B.A., Rakhimov, I.S.: Varieties of nilpotent complex Leibniz algebras of dimension less than five. Comm. Algebra 33(5), (2005) 3. Albeverio, S., Omirov, B.A., Rakhimov, I.S.: Classification of 4-dimensional nilpotent complex Leibniz algebras. Extracta Math. 21(3), (2006) 4. Ayupov, S.A., Omirov, B.A.: On Leibniz algebras. Algebra and Operator Theory (Tashkent, 1997), pp Kluwer Acad. Publ., Dordrecht (1997) 5. Ayupov, S.A., Omirov, B.A.: On a description of irreducible component in the set of nilpotent Leibniz algebras containing the algebra of maximal nilindex, and classification of graded filiform Leibniz algebras. In: Computer algebra in scientific computing (Samarkand, 2000), pp Springer, Berlin (2000) 6. Ayupov, S.A., Omirov, B.A.: On some classes of nilpotent Leibniz algebras. Siberian Math. J. 42(1), (2001) 7. Bergman, G.M.: The diamond lemma for ring theory. Adv. in Math. 29, (1978) 8. Cuvier, C.: Algèbres de Leibnitz: définitions, propriétés. Ann. Sci. École Norm. Sup. 27(1) (4), 1 45 (1994) 9. de Graaf, W.: Lie algebras: theory and algorithms. North-Holland Publishing Co., Amsterdam (2000)

10 186 J.M. Casas et al. 10. de Graaf, W.: Classification of solvable Lie algebras. Experiment. Math. 14, (2005) 11. Goze, M., Khakimdjanov, Y.: Nilpotent Lie algebras. Mathematics and its Applications, vol Kluwer Acad. Publ., Dordrecht (1996) 12. Helton, J.W., Miller, R.L., Stankus, M.: NCAlgebra: A Mathematica package for doing noncommuting algebra (1996), Insua, M.A.: Varias perspectivas sobre las bases de Gröbner: forma normal de Smith, algoritmo de Berlekamp y álgebras de Leibniz. Ph. D. Thesis (2005) 14. Insua, M.A., Ladra, M.: Gröbner bases in universal enveloping algebras of Leibniz algebras. J. Symbolic Comput. 44, (2009) 15. Jacobson, N.: Lie algebras. Interscience Publ., New York (1962) 16. Kurdiani, R.: Cohomology of Lie algebras in the tensor category of linear maps. Comm. Algebra 27(10), (1999) 17. Loday, J.-L.: Une version non commutative des algèbres de Lie: les algèbres de Leibniz. Enseign. Math. 39(2), (1993) 18. Loday, J.-L.: Algèbres ayant deux opérations associatives (digèbres). C. R. Acad. Sci. Paris Sér. I Math. 321(2), (1995) 19. Loday, J.-L., Pirashvili, T.: Universal enveloping algebras of Leibniz algebras and (co)homology. Math. Ann. 296, (1993) 20. Mora, T.: An introduction to commutative and noncommutative Gröbner bases. Theor. Comp. Sci. 134, (1994)